Particle System ignores vertex group
Please know Blender may include a face if even one vertex of a face is selected.  The whole face need not be selected.  This may be inconvenient for a brief moment.  The other hair placement tools are more precise.
If you use a mesh with more vertices and a group, then you may see this pattern above.  
Precise hair placement can be done with UV Maps and Particle Edit mode tools for hair.  See image below.




In the image above you see Particle Edit Mode offered. Weight Painting is also offered.  The yellow arrow points to 

Textures Panel associated with a Particle System while the Cycles Render is offered. Blender has 


Particle Edit offers the most precise placement on a 3D model
Textures can control density and length as an abbreviated explanation
Textures can have higher control Density than a low poly mesh
Vertex Groups can control density and length as an abbreviated explanation
Weight Painting a Vertex Group. Reasonable but least precise.




Consider

In the example below a placement of particles was desired on a single quad strip.  One solution was to duplicated the quad strip, place particles and move keep the meshes coincident.




Perhaps your vertex group is empty or has a weight close to 0.0 zero.
Perhaps you need to assign a weight of 1.0 to the group.